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The Skolem property in rings of integer-valued polynomials

Authors: Jean-Luc Chabert, Scott T. Chapman and William W. Smith
Journal: Proc. Amer. Math. Soc. 126 (1998), 3151-3159
MSC (1991): Primary 13B25; Secondary 11S05, 12J10, 13E05, 13G05
MathSciNet review: 1452796
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Abstract: Let $D$ be an integral domain with quotient field $K$ and $E\subseteq D$. We investigate the relationship between the Skolem and completely integrally closed properties in the ring of integer-valued polynomials \[ \mathrm {Int}(E, D) = \{f(X) \mid f(X) \in K[X] \text { and } f(a)\in D \text { for every } a\in E\}. \] Among other things, we show for the case $D=\mathbb {Z}$ and $\vert E \vert = \infty$ that the following are equivalent: (1) $\text {Int}(E,\mathbb {Z})$ is strongly Skolem, (2) $\text {Int}(E,\mathbb {Z})$ is completely integrally closed, and (3) $\text {Int}(E,\mathbb {Z})= \text {Int}(E\backslash \{a\}, \mathbb {Z})$ for every $a\in E$.

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Additional Information

Jean-Luc Chabert
Affiliation: Faculté de Mathématiques et d’Informatique, Université de Picardie, 33 rue Saint Leu, 80 039 Amiens, France

Scott T. Chapman
Affiliation: Department of Mathematics, Trinity University, 715 Stadium Drive, San Antonio, Texas 78212-7200
MR Author ID: 47470

William W. Smith
Affiliation: Department of Mathematics, The University of North Carolina at Chapel Hill, North Carolina 27599-3250

Keywords: Integer-valued polynomial, Skolem property, Prüfer domain
Received by editor(s): October 10, 1996
Received by editor(s) in revised form: March 25, 1997
Additional Notes: Part of this work was completed while the third author was on leave visiting Trinity University.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society